<h2>The ring of Siegel modular forms of degree 2 with respect to 
<a class="knowl-title" knowl="mf.siegel.group.gamma0">$\Gamma_0(4)$</a>
with and without
<a class="knowl-title" knowl="mf.siegel.character">character</a>
</h2>

<div class="literature">
  <ul>
    <li><span class="name">H. Aoki, T.Ibukiyama: </span>Simple graded rings of Siegel modular forms, differential operators and Borcherds products. Internat. J. Math. 16 (2005), 249-279, <a href="http://www.ams.org/mathscinet-getitem?mr==2130626">MR2130626</a></li>
    <li><span class="name">S. Hayashida, T.Ibukiyama: </span>Siegel modular forms of
half integral weight and a lifting conjecture.  J. Math. Kyoto Univ., (2004),  - , <a href="http://www.ams.org/mathscinet-getitem?mr==??">MR???</a></li>
  </ul>
</div>

<p>
  By the results in above papers,
  the ring <script type="math/tex">M_{*}(\Gamma_0(4))</script> of Siegel modular forms of degree 2 with respect to the group 
<a class="knowl-title" knowl="mf.siegel.group.gamma0">$\Gamma_0(4)$</a>
and the module
$\oplus_k M_{k}(\Gamma_0(4),\psi_4)$
with respect to the group 
<a class="knowl-title" knowl="mf.siegel.group.gamma0">$\Gamma_0(4)$</a>
and
<a class="knowl-title" knowl="mf.siegel.character">character</a>
$\psi_4$
is generated by the following generators, which involve the usage of 
<a class="knowl-title" knowl="mf.siegel.theta_constant">theta constants</a>.

<ul>
<li>
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Gamma0_4', form = 'Generator_I_2', weight = 2, page = 'specimen') }}">$X$</a>, a form of weight 2,
with formula
$$X = ((\theta_{0000})^4+(\theta_{0001})^4+(\theta_{0010})^4+(\theta_{0011})^4)/4.$$
</li>
<li>
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Gamma0_4', form = 'Generator_II_2', weight = 2, page = 'specimen') }}">$X(2\Omega)$</a>, a form  of weight 2.
</li>
<li>
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Gamma0_4', form = 'Generator_4', weight = 4, page = 'specimen') }}">$f_2(2\Omega)$</a>, a form  of weight 4,
with formula
$$f_2 = (\theta_{0000})^4.$$
</li>
<li>
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Gamma0_4', form = 'Generator_6', weight = 6, page = 'specimen') }}">$K(2\Omega)$</a>, a form  of weight 6,
with formula
$$K = (\theta_{0100}\theta_{0110}\theta_{1000}\theta_{1001}\theta_{1100}\theta_{1111})^2/4096.$$
</li>
<li>
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Gamma0_4', form = 'Generator_11', weight = 11, page = 'specimen') }}">$f_{11}(2\Omega)$</a>, a cusp form of weight 11,
with formula
$$f_{11} = f_6\chi_5,$$
where
<ul>
<li>
$\chi_5=\theta_{0000}\theta_{0001}\theta_{0010}\theta_{0011}\theta_{0100}\theta_{0110}\theta_{1000}\theta_{1001}\theta_{1100}\theta_{1111}$
</li>
<li>
$f_6 = ((\theta_{0001})^{4}-(\theta_{0010})^{4})
((\theta_{0001})^{4}-(\theta_{0011})^{4})((\theta_{0010})^{4}-(\theta_{0011})^{4})$.
</li>
</ul>
</li>
</ul>
Note that we write $F(2\Omega)$ to mean "apply $F$ after doubling the input".

The generators $X, X(2\Omega),f_2(2\Omega), K(2\Omega)$ are algebraically independent.
Denote $B={\Bbb C}[X, X(2\Omega),f_2(2\Omega), K(2\Omega)]$.
Then the ring of modular forms is

$$M(\Gamma_0(4)) =
B + Y(2\Omega) B + f_{11}(2\Omega)(B + Y(2\Omega) B),$$
where
$Y = (\theta_{0000}\theta_{0001}\theta_{0010}\theta_{0011})^2.$


The ideal of cusp forms is ..??...


<p>
We have the module of modular forms with character
of even weights only is
$$\oplus_{k=0}^\infty M_{2k}(\Gamma_0(4),\psi_4) =
f_{11}(2\Omega)f_{1}(2\Omega)B + Y(2\Omega) B + f_{11}(2\Omega)f_{3}(2\Omega)B,$$
</p>
where $f_1=\theta_{0000}^2$
and $f_3=(\theta_{0000}\theta_{0001}\theta_{0011})^2$.
